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Hilbert Functions and Pseudo‐Rational Local Rings of Dimension Two
Author(s) -
Rees D.
Publication year - 1981
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-24.3.467
Subject(s) - mathematics , local ring , residue field , pure mathematics , dimension (graph theory) , multiplicity (mathematics) , ideal (ethics) , maximal ideal , combinatorics , ring (chemistry) , discrete mathematics , field (mathematics) , mathematical analysis , law , chemistry , organic chemistry , political science
Let Q be an analytically unramified Cohen‐Macaulay local ring of dimension 2, with maximal ideal m and infinite residue field k . If a is an m‐primary ideal of Q , a ∗ will denote its integral closure, λ(a) = l( Q /a ∗ ), e (a) will denote its multiplicity, θ(a) = 2. λ(a)− e (a) and θ ¯ (a) = lim θ(a n )/ n . This paper uses explicit formulae for θ(a r ), θ(a r b s ) related to earlier results of Narita to prove that θ ¯ (ab) = θ ¯ (a) + θ ¯ (b), and that the identity θ(ab) = θ(a) + θ(b) holding for all m‐primary ideals a, b characterises the pseudo‐rational local rings of J. Lipman among normal Q . This is used to prove a number of results concerning the normal genus of an ideal and pseudo‐rational local rings. In the last section, an expression for θ(a) is obtained in the case where Q is regular which is related to the theory of infinitely near points of D. G. Northcott.